Find the mean, variance, and standard deviation for each of...

Find the mean, variance, and standard deviation for each of the values of n and p when the conditions for the binomial distribution are met. $$ \begin{array}{l}{\text { a. } n=100, p=0.75} \\ {\text { b. } n=300, p=0.3} \\ {\text { c. } n=20, p=0.5} \\ {\text { d. } n=10, p=0.8}\end{array} $$

Find the mean, variance, and standard deviation for
each of the values of n and p when the conditions for
the binomial distribution are met.
$$
\begin{array}{l}{\text { a. } n=100, p=0.75} \\ {\text { b. } n=300, p=0.3} \\ {\text { c. } n=20, p=0.5} \\ {\text { d. } n=10, p=0.8}\end{array}
$$

Key Concepts

Binomial Distribution

A binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and a constant probability of success. It is widely used in probabilistic and statistical analyses when the conditions for binomial trials are met.

Mean of a Binomial Distribution

The mean, or expected value, of a binomial distribution is calculated using the formula np, where n is the total number of trials and p is the probability of success in each trial. This value represents the average number of successes one can expect over many repetitions of the experiment.

Variance of a Binomial Distribution

The variance of a binomial distribution measures the spread or dispersion of the number of successes around the mean and is given by the formula np(1-p), where 1-p is the probability of failure. This calculation helps in understanding the degree of variability inherent in the process.

Standard Deviation of a Binomial Distribution

The standard deviation, which is the square root of the variance, provides a measure of how much the number of successes diverges from the mean on average. It is calculated as the square root of np(1-p) and offers insight into the reliability and predictability of the outcomes in binomial experiments.

Transcript

00:01 So let's look at finding mean variance in standard deviation of binomial distributions.

00:09 So to find the mean, we're going to use the symbol mu to represent mean.

00:14 And to find it, we multiply the number, which we like to call n times the probability of success, which we call p.

00:24 The variance, which will use this to symbolize variance, we like to do number.

00:31 Of trials, whatever number of people surveyed, whatever it is, the n, we multiply it by the probability of success and we multiply it by q.

00:43 What we know as q is the probability of failure or one minus the probability of success.

00:53 And to find the standard deviation, we will take the square root of the variance of n times p times q.

01:03 Or you might write that as the square root of the variance.

01:09 So let's take a look at some examples.

01:12 If n equals 100 and p equals 0 .75, what is the mean? the mean is 100 times 0 .75, which is 75.

01:28 The variance is 100 times 0 .75 times 0 .25, which is 18 .75.

01:42 And the standard deviation is the square root of 18 .75, which is about 4 .33.

01:56 What if n equals 300? p equals 0 .3? the mean, the mean, would be 300 times 0 .3, which is 90.

02:14 The variance would be 300 times 0 .3 times 0 .7.

02:23 Remember, to find this value of 0 .7, i just do 1 minus p, which is 63...

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